Exact Real Arithmetic - Computable reals. Inspired by ''The most unreliable technique for computing pi.'' See also http://www.haskell.org/haskellwiki/Exact_real_arithmetic .
- type T = (Exponent, Mantissa)
- type FixedPoint = (Integer, Mantissa)
- type Mantissa = [Digit]
- type Digit = Int
- type Exponent = Int
- type Basis = Int
- moveToZero :: Basis -> Digit -> (Digit, Digit)
- checkPosDigit :: Basis -> Digit -> Digit
- checkDigit :: Basis -> Digit -> Digit
- nonNegative :: Basis -> T -> T
- nonNegativeMant :: Basis -> Mantissa -> Mantissa
- compress :: Basis -> T -> T
- compressFirst :: Basis -> T -> T
- compressMant :: Basis -> Mantissa -> Mantissa
- compressSecondMant :: Basis -> Mantissa -> Mantissa
- prependDigit :: Basis -> T -> T
- trim :: T -> T
- trimUntil :: Exponent -> T -> T
- trimOnce :: T -> T
- decreaseExp :: Basis -> T -> T
- pumpFirst :: Basis -> Mantissa -> Mantissa
- decreaseExpFP :: Basis -> (Exponent, FixedPoint) -> (Exponent, FixedPoint)
- pumpFirstFP :: Basis -> FixedPoint -> FixedPoint
- negativeExp :: Basis -> T -> T
- fromBaseCardinal :: Basis -> Integer -> T
- fromBaseInteger :: Basis -> Integer -> T
- mantissaToNum :: C a => Basis -> Mantissa -> a
- mantissaFromCard :: C a => Basis -> a -> Mantissa
- mantissaFromInt :: C a => Basis -> a -> Mantissa
- mantissaFromFixedInt :: Basis -> Exponent -> Integer -> Mantissa
- fromBaseRational :: Basis -> Rational -> T
- toFixedPoint :: Basis -> T -> FixedPoint
- fromFixedPoint :: Basis -> FixedPoint -> T
- toDouble :: Basis -> T -> Double
- fromDouble :: Basis -> Double -> T
- fromDoubleApprox :: Basis -> Double -> T
- fromDoubleRough :: Basis -> Double -> T
- mantLengthDouble :: Basis -> Exponent
- liftDoubleApprox :: Basis -> (Double -> Double) -> T -> T
- liftDoubleRough :: Basis -> (Double -> Double) -> T -> T
- showDec :: Exponent -> T -> String
- showHex :: Exponent -> T -> String
- showBin :: Exponent -> T -> String
- showBasis :: Basis -> Exponent -> T -> String
- powerBasis :: Basis -> Exponent -> T -> T
- rootBasis :: Basis -> Exponent -> T -> T
- fromBasis :: Basis -> Basis -> T -> T
- fromBasisMant :: Basis -> Basis -> Mantissa -> Mantissa
- cmp :: Basis -> T -> T -> Ordering
- lessApprox :: Basis -> Exponent -> T -> T -> Bool
- trunc :: Exponent -> T -> T
- equalApprox :: Basis -> Exponent -> T -> T -> Bool
- ifLazy :: Basis -> Bool -> T -> T -> T
- mean2 :: Basis -> (Digit, Digit) -> (Digit, Digit) -> Digit
- withTwoMantissas :: Mantissa -> Mantissa -> a -> ((Digit, Mantissa) -> (Digit, Mantissa) -> a) -> a
- align :: Basis -> Exponent -> T -> T
- alignMant :: Basis -> Exponent -> T -> Mantissa
- absolute :: T -> T
- absMant :: Mantissa -> Mantissa
- fromLaurent :: T Int -> T
- toLaurent :: T -> T Int
- liftLaurent2 :: (T Int -> T Int -> T Int) -> T -> T -> T
- liftLaurentMany :: ([T Int] -> T Int) -> [T] -> T
- add :: Basis -> T -> T -> T
- sub :: Basis -> T -> T -> T
- neg :: Basis -> T -> T
- addSome :: Basis -> [T] -> T
- addMany :: Basis -> [T] -> T
- type Series = [(Exponent, T)]
- series :: Basis -> Series -> T
- seriesPlain :: Basis -> Series -> T
- splitAtPadZero :: Int -> Mantissa -> (Mantissa, Mantissa)
- splitAtMatchPadZero :: [()] -> Mantissa -> (Mantissa, Mantissa)
- truncSeriesSummands :: Series -> Series
- scale :: Basis -> Digit -> T -> T
- scaleSimple :: Basis -> T -> T
- scaleMant :: Basis -> Digit -> Mantissa -> Mantissa
- mulSeries :: Basis -> T -> T -> Series
- mul :: Basis -> T -> T -> T
- divide :: Basis -> T -> T -> T
- divMant :: Basis -> Mantissa -> Mantissa -> Mantissa
- divMantSlow :: Basis -> Mantissa -> Mantissa -> Mantissa
- reciprocal :: Basis -> T -> T
- divIntMant :: Basis -> Int -> Mantissa -> Mantissa
- divIntMantInf :: Basis -> Int -> Mantissa -> Mantissa
- divInt :: Basis -> Digit -> T -> T
- sqrt :: Basis -> T -> T
- sqrtDriver :: Basis -> (Basis -> FixedPoint -> Mantissa) -> T -> T
- sqrtMant :: Basis -> Mantissa -> Mantissa
- sqrtFP :: Basis -> FixedPoint -> Mantissa
- sqrtNewton :: Basis -> T -> T
- sqrtFPNewton :: Basis -> FixedPoint -> Mantissa
- lazyInits :: [a] -> [[a]]
- expSeries :: Basis -> T -> Series
- expSmall :: Basis -> T -> T
- expSeriesLazy :: Basis -> T -> Series
- expSmallLazy :: Basis -> T -> T
- exp :: Basis -> T -> T
- intPower :: Basis -> Integer -> T -> T -> T -> T
- cardPower :: Basis -> Integer -> T -> T -> T
- powerSeries :: Basis -> Rational -> T -> Series
- powerSmall :: Basis -> Rational -> T -> T
- power :: Basis -> Rational -> T -> T
- root :: Basis -> Integer -> T -> T
- cosSinhSmall :: Basis -> T -> (T, T)
- cosSinSmall :: Basis -> T -> (T, T)
- cosSinFourth :: Basis -> T -> (T, T)
- cosSin :: Basis -> T -> (T, T)
- tan :: Basis -> T -> T
- cot :: Basis -> T -> T
- lnSeries :: Basis -> T -> Series
- lnSmall :: Basis -> T -> T
- lnNewton :: Basis -> T -> T
- lnNewton' :: Basis -> T -> T
- ln :: Basis -> T -> T
- angle :: Basis -> (T, T) -> T
- arctanSeries :: Basis -> T -> Series
- arctanSmall :: Basis -> T -> T
- arctanStem :: Basis -> Int -> T
- arctan :: Basis -> T -> T
- arctanClassic :: Basis -> T -> T
- zero :: T
- one :: T
- minusOne :: T
- eConst :: Basis -> T
- recipEConst :: Basis -> T
- piConst :: Basis -> T
- sliceVertPair :: [a] -> [(a, a)]
Converts all digits to non-negative digits, that is the usual positional representation. However the conversion will fail when the remaining digits are all zero. (This cannot be improved!)
Requires, that no digit is
The leading digit might be negative and might be
Compress second digit. Sometimes this is enough to keep the digits in the admissible range. Does not prepend a digit.
Accept a high leading digit for the sake of a reduced exponent.
This eliminates one leading digit.
pumpFirst but with exponent management.
Merge leading and second digit.
This is somehow an inverse of
Make sure that a number with absolute value less than 1 has a (small) negative exponent. Also works with zero because it chooses an heuristic exponent for stopping.
Only return as much digits as are contained in Double. This will speedup further computations.
Convert from a
b basis representation to a
Works well with every exponent.
Convert from a
b^e basis representation to a
Works well with every exponent.
Convert between arbitrary bases. This conversion is expensive (quadratic time).
The basis must be at least ***. Note: Equality cannot be asserted in finite time on infinite precise numbers. If you want to assert, that a number is below a certain threshold, you should not call this routine directly, because it will fail on equality. Better round the numbers before comparison.
If all values are completely defined, then it holds
if b then x else y == ifLazy b x y
b is undefined,
then it is at least known that the result is between
mean2 b (x0,x1) (y0,y1)
round ((x0.x1 + y0.y1)/2) ,
y0.y1 are positional rational numbers
with respect to basis
Get the mantissa in such a form that it fits an expected exponent.
(e, alignMant b e x) represent the same number.
Add at most
More summands will violate the allowed digit range.
Add many numbers efficiently by computing sums of sub lists with only little carry propagation.
Add an infinite number of numbers. You must provide a list of estimate of the current remainders. The estimates must be given as exponents of the remainder. If such an exponent is too small, the summation will be aborted. If exponents are too big, computation will become inefficient.
but it pads with zeros if the list is too short.
This way it preserves
length (fst (splitAtPadZero n xs)) == n
Undefined if the divisor is zero - of course. Because it is impossible to assert that a real is zero, the routine will not throw an error in general.
ToDo: Rigorously derive the minimal required magnitude of the leading divisor digit.
Fast division for small integral divisors, which occur for instance in summands of power series.
We need a leading digit of type Integer,
because we have to collect up to 4 digits.
This presentation can also be considered as
ToDo: Rigorously derive the minimal required magnitude of the leading digit of the root.
nth digit of the square root
depends roughly only on the first
n digits of the input.
This is because
sqrt (1+eps) .
However this implementation requires
equalApprox 1 + eps/2
2*n input digits
This is due to the repeated use of
It would suffice to fully compress only every
basisth iteration (digit)
and compress only the second leading digit in each iteration.
Can the involved operations be made lazy enough to solve
y = (x+frac)^2
frac = (y-x^2-frac^2) / (2*x) ?
Newton iteration doubles the number of correct digits in every step. Thus we process the data in chunks of sizes of powers of two. This way we get fastest computation possible with Newton but also more dependencies on input than necessary. The question arises whether this implementation still fits the needs of computational reals. The input is requested as larger and larger chunks, and the input itself might be computed this way, e.g. a repeated square root. Requesting one digit too much, requires the double amount of work for the input computation, which in turn multiplies time consumption by a factor of four, and so on.
Optimal fast implementation of one routine does not preserve fast computation of composed computations.
The routine assumes, that the integer parts is at least
List.inits is defined by
inits = foldr (x ys ->  : map (x:) ys) []
This is too strict for our application.
Prelude> List.inits (0:1:2:undefined) [,,[0,1]*** Exception: Prelude.undefined
The following routine is more lazy than
and even lazier than
but it is restricted to infinite lists.
This degree of laziness is needed for
Prelude> lazyInits (0:1:2:undefined) [,,[0,1],[0,1,2],[0,1,2,*** Exception: Prelude.undefined
Residue estimates will only hold for exponents with absolute value below one.
The computation is based on
thus the denominator should not be too big.
(Say, at most 1000 for 1000000 digits.)
It is not optimal to split the power into pure root and pure power (that means, with integer exponents). The root series can nicely handle all exponents, but for exponents above 1 the series summands rises at the beginning and thus make the residue estimate complicated. For powers with integer exponents the root series turns into the binomial formula, which is just a complicated way to compute a power which can also be determined by simple multiplication.
cosSinSmall but converges faster.
cosSinSmall with reduced arguments
using the trigonometric identities
cos (4*x) = 8 * cos x ^ 2 * (cos x ^ 2 - 1) + 1
sin (4*x) = 4 * sin x * cos x * (1 - 2 * sin x ^ 2)
Note that the faster convergence is hidden by the overhead.
The same could be achieved with a fourth power of a complex number.
x' = x - (exp x - y) / exp x = x + (y * exp (-x) - 1)
First, the dependencies on low-significant places are currently
much more than mathematically necessary.
*Number.Positional> expSmall 1000 (-1,100 : replicate 16 0 ++ [undefined])
(0,[1,105,171,-82,76*** Exception: Prelude.undefined
Every multiplication cut off two trailing digits.
*Number.Positional> nest 8 (mul 1000 (-1,repeat 1)) (-1,100 : replicate 16 0 ++ [undefined])
(-9,[101,*** Exception: Prelude.undefined
Possibly the dependencies of expSmall
could be resolved by not computing
mul series which are merged and subsequently added.
But this would lead to an explosion of series.
Second, even if the dependencies of all atomic operations
are reduced to a minimum,
the mathematical dependencies of the whole iteration function
are less than the sums of the parts.
Lets demonstrate this with the square root iteration.
(1.4140 + 21.4140) 2 == 1.414213578500707
(1.4149 + 21.4149) 2 == 1.4142137288854335
That is, the digits
213 do not depend mathematically on
but their computation depends.
Maybe there is a glorious trick to reduce the computational dependencies
to the mathematical ones.
This is an inverse of
also known as
atan2 with flipped arguments.
It's very slow because of the computation of sinus and cosinus.
However, because it uses the
atan2 implementation as estimator,
the final application of arctan series should converge rapidly.
It could be certainly accelerated by not using cosSin and its fiddling with pi. Instead we could analyse quadrants before calling atan2, then calling cosSinSmall immediately.
Arcus tangens of arguments with absolute value less than
1 / sqrt 3.
Efficient computation of Arcus tangens of an argument of the form
This implementation gets the first decimal place for free
by calling the arcus tangens implementation for
A classic implementation without ''cheating'' with floating point implementations.
x < 1 / sqrt 3
1 / sqrt 3 == tan (pi/6))
arctan power series.
sqrt 3 is approximately
x > sqrt 3
arctan x = pi/2 - arctan (1/x)
arctan x = pi/4 - 0.5*arctan ((1-x^2)/2*x)
(which follows from
arctan x + arctan y == arctan ((x+y) / (1-x*y))
which in turn follows from complex multiplication
(1:+x)*(1:+y) == ((1-x*y):+(x+y))
x is close to
sqrt 3 or
1 / sqrt 3 the computation is quite inefficient.